Method for the control of a harmonically oscillating load

ABSTRACT

The object of the invention is a method to control a harmonically oscillating load, in which method the load is transferred from a beginning state to a final state of the load oscillation and to a final velocity (V ref ) of the point of suspension by controlling the load with control sequences which comprise consecutive acceleration pulses, whereby the beginning and final states of load oscillation and the beginning and final velocities of the point of suspension are measured or estimated. The control sequence a(t) of the load is formed from many standard duration acceleration pulses (a 1 , a 2 , a 3 ) calculated on the basis of random beginning and final states of the load.

The object of the invention is a method for the control of aharmonically oscillating load, in which method the load is transferredfrom the initial state to the final state of the oscillation of the loadand to the final velocity of the point of suspension by controlling theload with control sequences, which consist of consecutively performedacceleration pulses, in which method the beginning or initial states ofthe oscillation of the load and the beginning or initial velocity of thepoint of suspension are either measured or estimated, and in whichmethod the desired final states of oscillation of the load and thedesired final velocity of the point of suspension are set by a humanoperator or, in the case of an automatic crane, by a computer.

A method is already known from the publication "Suboptimal control ofthe roof crane by using the microcomputer," by S. Yamada, H. Fujikawa,K. Matsumoto, IEEE CH1897-8/83/0000-0323, pp 323-328, in which, atconstant acceleration, variable acceleration and switching times arepre-calculated and tabulated for the load suspending apparatus so that,by using the acceleration and switching times, the velocity of the loadsuspending apparatus, the oscillating angle of the suspended load, andthe angular velocity of the oscillation of the suspended load aresteered from certain starting values to desired final values. In thismethod, the phase plane is divided into squares, and switching times foracceleration are calculated and entered for each phase plane square.Consequently, the system moves at the desired final velocity and thesuspended load is in a stationary state. The method uses constantacceleration, and the acceleration switching times are adjusted toachieve the desired final result. When using this method, if allpossible starting and finishing situations are allowed, the table willbe extremely large. In the method presented in the publication inquestion, the acceleration pulses are, in terms of absolute value,constantly large or at the value zero. In addition, the duration of theacceleration pulses is calculated iteratively and not directly bycalculation.

Furthermore, the magnitude of acceleration is partly determined so that,for example, the first acceleration pulse is, in terms of magnitude, thesame as the third acceleration pulse. Considering the phase planepresentation, in the above publication the position of the centre pointof the trajectory of the acceleration pulse is determined, but thelength of the curve varies.

In addition a method is known which, by summing theoscillation-eliminating acceleration sequences known from patentpublication U.S. Pat. No. 3,517,830, succeeds in forming a velocityinstruction for the load suspending apparatus which guides the loadsuspending apparatus to the desired final velocity in such a way thatthe oscillating angle of the suspended load is zero and the angularvelocity of the suspended load's oscillation is zero. The use of thismethod is limited by the requirement for a stationary beginningsituation of the suspended load and by the need to achieve a particularfinal situation concerning the oscillating angle and angular velocity ofthe suspended load. The method is not therefore suited to the guiding ofa suspended load's suspension apparatus and the angular velocity of thesuspended load to desired, random situations from completely randombeginning values. The solution presented in publication U.S. Pat. No.3,517,830 requires stationary beginning and final situations and itsdisadvantage is that it does not allow control adjustments inmid-control, but the sequence must be carried through to the end.

The disadvantage of the known methods is that they do not offer asimple, calculationally-advantageous method of calculating the velocityinstructions for a suspended load's suspension apparatus, which wouldguide the load's suspension apparatus to a desired random finalvelocity, and suspended load oscillating angle, and angular velocity ofthe suspended load oscillating angle, starting from random beginningvalues.

The purpose of the invention is to solve the problems described above.The said problem is solved by a method in accordance with the inventionnow presented, characterized by the fact that the load control sequenceis formed from a plurality of acceleration pulses, each pulse having aconstant rate of acceleration. The load control sequence provides ageneral solution for controlling a harmonically oscillating load for anyarbitrary initial or desired final value of swing angle, angular speed,and velocity of the point of suspension. Considering the phase planerepresentation, in the solution according to the invention, the locationof the centre point of the acceleration pulse trajectory varies with thevariation in the acceleration pulse value or magnitude, but the lengthof the trajectory curve is stable or at least pre-determined.

The invention offers a calculationally-advantageous way to define thesuspended load's acceleration and deceleration so that, starting fromany beginning velocity of the point of suspension of the suspended load,any oscillating angle of the suspended load and any angular velocity ofthe oscillating angle of the suspended load, it is possible to finishwith any velocity of the point of suspension of the suspended load, anyoscillating angle of the suspended load and any angular velocity of theoscillating angle of the suspended load in a desired, pre-determinedtime. Furthermore, the invention is calculationally-advantageous becauseit uses a previously set number of acceleration pulses of a previouslyset duration, and because the magnitude of the pulses are easilycalculated using simple equations based on the initial and desired finalvalues of swing angle, angular speed, and velocity of the point ofsuspension. The invention may be utilized in the control of allsuspension systems where, due to the method of suspension, there isharmonic oscillation of the load. The invention is adaptable, forexample, to overhead cranes.

The developed method is especially suitable for use in equipment wherethe position of the suspended load is measured. Then with the method itis possible to rapidly calculate the control for guiding the load to thedesired position and velocity. In systems in which the position of theload is not measured, the beginning values for the oscillating angle andthe angular velocity of the load are estimated using a mathematicalmodel.

In the following, the invention is explained with reference to theattached figures, in which

FIG. 1 shows the principles of harmonic oscillation,

FIG. 2a shows a velocity sequence known per se,

FIG. 2b shows a phase plane representation corresponding to FIG. 2a,

FIG. 3a shows another velocity sequence known per se,

FIG. 3b shows a phase plane representation corresponding to FIG. 3a,

FIG. 4 shows a phase plane representation,

FIG. 5 shows the velocity and acceleration coefficient,

FIG. 6 shows a phase plane representation corresponding to FIG. 5,

FIG. 7 shows a flow chart of the method according to the invention.

With reference to FIG. 1 the following symbols are shown:

X_(r) =the position of the suspension point in the x direction

X_(t) =the position of the suspended load in the x direction

Y_(t) =the position of the suspended load in the y direction

l=the length of the suspension rope of the load

g=the gravitational acceleration

m=the mass of the load

The position of the suspended load is obtained from FIG. 1 by theequations (1) and (2).

    X.sub.t =X.sub.r -l·sin Θ                   (1)

    Y.sub.t =l·sin θ                            (2)

The kinetic energy of the load W is obtained from the formula (3).##EQU1##

By combining equations (1) and (2) with equation (3) we obtain thekinetic energy of the suspended load on the polar coordinates (4).##EQU2##

The potential energy of the load is obtained from FIG. 1 by equation(5).

    V=-mgl cos θ                                         (5)

As is known, the Lagrange function is

    L=W-V.                                                     (6)

By combining equations (4) and (5) with equation (6), the Lagrangefunction in this case is equation (7). ##EQU3##

The system's equation of motion is derived from the Lagrange function Lby combining it with the Lagrange equation of motion (8). ##EQU4## whereL=the Lagrange function

q_(i) =the i:th coordinate

Q_(i) =the force having effect from outside the system.

By combining the derived Lagrange function (7) with the Lagrangeequation of motion (8) and by performing the derivations we obtain theequation (9) as the system's equation of motion. ##EQU5##

With a small oscillating angle (Θ<10°) sin Θ≈0 and cos Θ≈1.

With these assumptions the equation (9) simplifies to the form (10).##EQU6##

It can be seen from equation (10) that the oscillating angle Θ of thesuspended load is controlled by the acceleration of the load'ssuspension point x_(r). The phase plane representation can be achievedfrom the equation (10) by multiplying the equation with dΘ/dt to achieveequation (11). ##EQU7##

According to the rule ##EQU8##

the equation (13) is obtained from the equation (11). ##EQU9##

By integrating the equation (13) in the Θ ratio, (14) is obtained.##EQU10##

When we assume that the system's beginning situation is a stationarystate (t=0, Θ=0, dΘ/dt=0) the integration constant C is zero. Thus theequation (15) is obtained. ##EQU11##

By entering a for the acceleration of the point of suspension of theload, equation (16) is obtained from equation (15). ##EQU12## and againby entering ##EQU13## we obtain equation (18). ##EQU14##

It can be seen from equation (18) that the load oscillation plotsconcentric circles in the phase plane (O,a/g). FIGS. 2a, 2b, 3a and 3billustrate the phase plane representation. FIGS. 2a and 2b show a cranevelocity sequence known per se and the corresponding phase planerepresentation. In the case of FIGS. 2a and 2b the system is acceleratedat an even acceleration a by the time τ corresponding to the system'scharacteristic oscillation time. The characteristic oscillation time forthe mathematical pendulum is obtained from the formula ##EQU15##

It can be seen from the phase plane that a concentric circle (O,a/g) isthen plotted on the angle/angular velocity-coordinates. In FIGS. 3a and3b the system is accelerated at sequences which comprise two constantacceleration pulses of length τ/6 and a period of steady velocity oflength τ/3. The system was at the beginning situation in a stationarystate of rest, so that the load's oscillating angle and angular velocityare zero. When the system is accelerated at an even acceleration a, aconcentric circle (O, a/g) is plotted on the phase plane, which touchesthe beginning point (0.0). When in FIGS. 2a and 2b the system isaccelerated at an even acceleration a and time τ (characteristicoscillation time), a complete circle is plotted on the phase plane. InFIGS. 3a and 3b the length of the first acceleration pulse is τ/6, whena concentric circular arc (O, a/g) is plotted on the phase planestarting from the point (0.0), with the length of the arc being 360/6=60degrees. The next step in the velocity sequence is a phase of evenvelocity, when the system's acceleration a=0. Then a concentric circulararc (0.0) is plotted on the phase plane starting from the point in thephase plane at which the previous acceleration sequence finished. As thelength of the even velocity phase is τ/3, a concentric arc whose lengthis 360/3=120 degrees is plotted on the phase plane. Finally the systemis accelerated again at an acceleration a and time (τ/6). Then aconcentric arc (O,a/g) is again plotted on the phase plane, whose archas a length of 360/6=60 degrees and which begins at the point where theprevious acceleration pulse (constant velocity a=0) finished. It can beseen in FIG. 3b that the system states ended as zero after a period τ/6.If the system's acceleration a continues to be zero, the system ismoving at a constant velocity without load oscillation.

A harmonically oscillating load 3, for example on an overhead crane, istransferred from the beginning state to the final state of the load'soscillation and to the final velocity V_(ref) of the point of suspensionby controlling the load with a control sequence a(t), which comprisesconsecutively performed acceleration pulses a_(i). In the method thebeginning states of the load's oscillation and the beginning velocity ofthe point of suspension are measured or estimated. According to theinvention the load's control sequence a(t) is formed from accelerationpulses having a constant duration and a constant rate of acceleration(a₁, a₂, a₃ . . . a_(n)) The control sequence a(t) selects accelerationpulses having a preset duration, a preset constant rate of acceleration,and a calculated magnitude based on any measured or estimated beginningvalue for the load's oscillating motion, on any desired finishing valuefor the load's oscillating motion, on any beginning velocity of thepoint of suspension, and on any desired finishing velocity of the pointof suspension.

FIG. 4 also shows the phase plane formulas. With reference to FIG. 4,one of the calculationally-advantageous applications of the methodaccording to the invention is a control which leads to the desiredsystem final velocity, the desired oscillating angle and the desiredfinal velocity of the load's angle of oscillation, by adapting threeacceleration periods (a₁, a₂ and a₃) of length τ/4 so that they performthe desired change in system velocity Δ/v or dv. ##EQU16##

Because in a certain application of the method τ/4 has been chosen asthe i length of each acceleration period, each acceleration periodcorresponds to a circular arc (360/4=90) of 90 degrees covered in thephase plane, where the arc's centre point is (0,a_(i) /g), and thecircular arc's beginning point is (ω₁, Θ₁) and the finishing point is(ω₂, Θ₂). When this acceleration period has ended, the system state hastransferred from the point (ω₁, Θ₁) to the point (ω₂, Θ₂). Because thelength of the acceleration period was chosen as τ/4, the point (ω₂, Θ₂)can be calculated when in addition the acceleration a₁ is known fromformulas (21) and (22). ##EQU17##

In a certain application of the method a control is calculated whichimplements the desired change Δv of velocity of the point of suspensionand after which the load's oscillating angle and angular velocity havetransferred from the point (ω₀, Θ₀) of the phase plane to the point (ω₃,Θ₃) so that three periods a₁, a₂, a₃ of even acceleration and of lengthτ/4 are used. Accelerations a₁, a₂, a₃ may be solved by the equations(23)-(29). ##EQU18##

Of the variables of the equations (23)-(29), Δv, ω₀, Θ₀, ω₃, Θ₃ areknown. The accelerations a₁, a₂, a₃ of the equations are solved so thatthe unknown variables ω₁, Θ₁, ω₂, Θ₂ are reduced away from the finalequations. Thus for the accelerations a₁, a₂, a₃ the equations (30)-(32)are solved on the phase plane. ##EQU19##

As an example we calculate the accelerations a₁, a₂, a₃ which guide thecrane system from starting states X₀ =ω₀ =0.02 rad/2, Y₀ =Θ₀ =0.02 radto the final states X₃ =ω₃ =0.0 rad/2, Y₃ =Θ₃ =0.0 rad, so that thevelocity of the point of suspension changes from the beginning value 0.1m/s to the final value 0.5 m/s, when the load's lifting height 1=10 m.##EQU20##

The magnitudes of the accelerations a_(i) are defined therefore byapplying circular arcs, revolving anti-clockwise, to the phase plane,where the second coordinate of the centre point of the circles is a_(i)/g.

FIGS. 5 and 6 show a velocity and acceleration sequence and acorresponding phase plane formula for the case presented above. It canbe observed from FIG. 5 that the acceleration sequence a(t) comprisesthree parts, whose magnitudes are as large as those calculated above,i.e. a₁ /g=-0.0036, a₂ /g=0.2 and a₃ /g=-0.0036. Correspondingly in thephase plane we transfer anti-clockwise from the beginning point A (0.02,0.02) via points B and C to the origin 0.

The harmonic oscillator presented in FIG. 1 may be for example anoverhead crane which has a crane carriage 1 from which, by means of asuspension apparatus 2, a load 3 is suspended. The crane also has acontrol terminal 4 and control unit 5. The crane operator gives velocityinstructions V_(ref) from the control terminal which are directed viathe control unit to the crane, or in practice to the crab traversingmotors of the crane carriage 1. FIG. 7 shows a flow chart of the methodaccording to the invention, but FIG. 7 can also be regarded as aninternal block diagram of the control unit. With reference to FIGS. 1and 7, the velocity instruction V_(ref) given by the crane operator isread into the control unit 5 in the first block 101. In the next block,i.e. in the first testing block 102, the velocity instruction given bythe operator is compared with the previous velocity instruction and, ifit has changed, then in the next block 103 the oscillating angle Θ₀ ofthe load 3 and the load's angular velocity ω₀, which represent thebeginning situation, are read into the control unit. In addition, inblock 103 the desired velocity change dv is calculated. In the followingblock 104, standard duration (preferably τ/4) new controls oracceleration pulses a₁, a₂, a₃ are calculated on the basis of theequations (30)-(32) presented above and are entered in a specialprogramme performance table. In calculating the acceleration pulses wealso utilize the desired final states, in other words the angularvelocity ω and oscillating angle Θ of the load's final state.

In the following phase 106, after the second testing block 105, a newvelocity instruction is calculated from the entered acceleration pulsesa₁, a₂, a₃, which in the last block 107 is directed as an instruction tothe crane's crab traversing motors. If it is noticed in the firsttesting block 102 that the velocity instruction V_(ref) has not changedand if it is noticed in block 105 that the performance table is empty,then the velocity instruction V_(ref) given by the operator is takendirectly as the velocity in block 108, and is directed to the crane'scrab traversing motors in accordance with block 107.

In FIG. 1 the random beginning states of the load, i.e. the oscillatingangle Θ₀ of the load 3 and the load's angular velocity ω₀ and the load'svelocity v are obtained from the feedback lines 10-12. The desired finalstates, i.e. the oscillating angle Θ₁, of the load's final state, theangular velocity ω₁ and the velocity instruction V_(ref) are obtainedfrom the control lines 13-15. The velocity change dv is obtained fromthe difference of lines 15 and 12.

The new velocity instruction obtained from the acceleration pulses a₁,a₂, a₃, calculated in the way according to the invention, is directed asa control to the crane's crab traversing motors via the control line120.

In the method according to the invention, the magnitudes of the standardduration acceleration pulses are calculated on the basis of the desiredvelocity change dv of the point of suspension, as well as on the desiredbeginning and final values of the oscillating angle and the chosenduration time τ/n of the acceleration pulse. The value n is preferably4, and this trigonometrically produces the best and most simple resultin calculation from the point of view of the sine and cosine terms. Inthe method according to the invention the duration and switching timesof the acceleration pulses performed at constant acceleration arepredetermined.

Formulas (30)-(32) determine the magnitude of each standard durationacceleration pulse as a function of any arbitrary beginning andfinishing state (the load's oscillating angle Θ, the angular velocity ω,the load's final velocity). Each acceleration pulse a₁, a₂, a₃ is solveddirectly by calculation, not therefore by iteration. In the method'sembodiment, which is advantageous both in terms of calculation and theequipment solution, each acceleration pulse a₁, a₂, a₃ of the controlsequence a(t) is calculated from a standard duration calculationalapproximation as presented by formulas (30)-(32). In that case,therefore, the constant duration parts or at least the parts ofpredetermined length of the acceleration sequence a_(i) fulfilling thedesired velocity change dv, in other words the acceleration pulses a₁,a₂, a₃ are each directly formed or calculated as a function of the loadoscillation's random beginning and finishing states X₀, Y₀, X₃, Y₃(where x stands for angular velocity ω and y stands for the oscillatingangle Θ), and further as a function of the desired velocity change Δv ordv and the chosen individual acceleration pulse duration, which ispreferably τ/4, and further as a function of the gravitationalacceleration g. In addition to the above, a preferable embodiment whichimproves the practicability of the method is that the approximations ofthe acceleration pulses are chosen so that, if the calculational factorsto be used in forming each individual acceleration pulse a₁, a₂, a₃ soallow, the standard duration acceleration pulses and/or the accelerationpulses of predetermined length are formed to differ from each other inabsolute value. The formation, i.e. calculation, of the magnitude of theacceleration pulses is therefore free of mutual initial settings whichwould restrict the application of the method.

One possible application for the invention may be a crane system inwhich the load's oscillating angle and angular velocity, and thevelocity of the load's point of suspension may be freely controlled. Inthis case it is possible with the method according to the invention tocalculate a control where the final result is that the load's velocity,oscillating angle and angular velocity are the desired values. Forexample, if the crane is stopped, but the load oscillates and theoscillating angle and the angular velocity can be measured or perfectlymodelled with a mathematical model or simulator, it is possible with themethod according to the invention to calculate the acceleration pulseswhose number and duration are predetermined and after the performance ofwhich the crane moves at the desired final velocity without oscillationof the load.

In a certain application it is possible to read from the operator'scontrol terminal 4 the desired motion velocity V_(ref) of the crane,i.e. the velocity at which the crane and the load 3 should move withoutload oscillation so that the load's oscillating angle and angularvelocity are zero. In this application the load's oscillating angle andangular velocity are measured and the velocity is assumed to follow thedesired velocity request of the control system exactly. When thevelocity request given by the operator is changed, the load'soscillating angle, angular velocity and the velocities of the point ofsuspension are read at that moment, as well as the new desirednon-oscillating, final velocity of the crane and load. These values areinserted in the formulas according to the invention, and calculationsare made of the acceleration pulses, at the end of which the desiredfinal velocity without load oscillation is achieved.

In a certain application of the invention, the load's oscillating angleis measured, and the velocity of the load's point of suspension followsexactly the velocity instruction of the control system. In thisapplication, the dynamic model of the oscillation of the crane's load isexploited in the calculation of the angular velocity of loadoscillation.

In a certain application of the invention the velocity of the point ofsuspension of the load follows exactly the velocity instruction given bythe control system, and the load's oscillating angle or angular velocityis not measured, but the load's oscillating angle and angular velocityis assumed to behave according to a mathematical model or simulatordescribing the crane's dynamics.

In a certain application of the method according to the invention, theload's oscillating angle decreases evenly, whereupon the load'soscillating angle and angular velocity plot a spiral instead of a circleon the phase plane. This is taken into account in formulating theequations according to the invention so that the angular-angularvelocity point is approached in a certain relationship to the centrepoint of the circular motion per each length unit of the arc moving inthe circumference. It is a linear change which is reflected in theequations only as a coefficient and does not influence the solvabilityof the equations.

Although the invention is further explained in the examples given in theattached diagrams, it is clear that the invention is not limited only tothese. It may be adapted on demand within the framework of the inventionideas here presented.

I claim:
 1. A method for the control of a harmonically oscillating load,in which the load is transferred from a beginning state of oscillatingmovement of the load (θ₀, ω₀) and from a beginning velocity of a pointof suspension of the load (V₀) to a desired final state of oscillatingmovement of the load (θ_(ref), ω_(ref)) and to a desired final velocity(V_(ref)) of the point of suspension of the load, comprising the stepsof:(a) determining the beginning state of oscillating movement of theload (θ₀, ω₀) and the beginning velocity of the point of suspension ofthe load (V₀); (b) determining the desired final state of oscillatingmovement of the load (θ_(ref), ω_(ref)) and the desired final velocityof the point of suspension of the load (V_(ref)); and (c) controllingthe load with a control sequence (a(t)) comprising consecutivelyperformed acceleration pulses (a_(i)), wherein the control sequence a(t)is formed from a plurality of acceleration pulses (a₁, a₂, a₃ . . .a_(n)) having a constant rate of acceleration of a calculated magnitudeand a predetermined duration, wherein the formed control sequence isbased on the beginning and the desired final states of the oscillatingmovement of the load and on the beginning and the desired final velocityof the point of suspension of the load.
 2. A method according to claim1, further comprising the steps of determining a characteristicoscillation time (τ) for the load, wherein the predetermined duration ofeach of the plurality of acceleration pulses is τ/4, wherein themagnitudes of the acceleration pulses (a₁, a₂, a₃ . . . a_(n)) arecalculated based on the desired velocity change (V_(ref) -V₀) of thepoint of suspension, the beginning and desired final values (θ₀,θ_(ref)) of the oscillating angle, the beginning and desired finalvalues (ω₀, ω_(ref)) of the angular velocity, and the predeterminedduration (τ/4) of each acceleration pulse.
 3. A method according toclaim 2, wherein the beginning value of the load's oscillating angle(θ₀) and of the angular velocity (ω₀) are determined by a simulatordescribing the dynamics of a harmonic oscillator.
 4. A method accordingto claim 2, further comprising the steps of entering the plurality ofacceleration pulses (a₁, a₂, a₃, . . . a_(n)) in a program performancetable, testing contents of the program performance table, andconsecutively performing the acceleration pulses as the control sequenceto be implemented for a set velocity change (V_(ref) -V₀).
 5. A methodaccording to claim 1, wherein the constant rate of acceleration of eachacceleration pulse (a₁, a₂, a₃ . . . a_(n)) of the control sequence a(t)is determined via a calculational approximation.
 6. A method accordingto claim 5, wherein the magnitude of the constant rate of accelerationdiffers in absolute value among the acceleration pulses.
 7. A methodaccording to claim 1, wherein the control sequence (a(t)) comprisesthree acceleration pulses (a₁, a₂, a₃), wherein each of the threeacceleration pulses have a duration of τ/4.
 8. A method according toclaim 1, wherein the beginning states of load oscillation and thebeginning velocity of the point of suspension are measured.
 9. A methodaccording to claim 1, wherein the beginning states of load oscillationand the beginning velocity of the point of suspension are estimated. 10.A method for the control of a harmonically oscillating load, in whichthe load is transferred from a beginning state of oscillating movementof the load (θ₀, ω₀) and from a beginning velocity of a point ofsuspension (V₀) to a desired final state of oscillating movement of theload (θ_(ref), ω_(ref)) and to a desired final velocity (V_(ref)) of thepoint of suspension, comprising the steps of:(a) determining thebeginning states of oscillating movement of the load (θ₀, ω₀) and thebeginning velocity of the point of suspension of the load (V₀); (b)determining the desired final states of oscillating movement of the load(θ_(ref), ω_(ref)) and the desired final velocity of the point ofsuspension of the load (V_(ref)); and (c) controlling the load with acontrol sequence (a(t)) comprising consecutively performed accelerationpulses (a_(i)), wherein the control sequence a(t) is formed from aplurality of acceleration pulses (a₁, a₂, a₃ . . . a_(n)) having aconstant rate of acceleration of a calculated magnitude and apredetermined duration, wherein the magnitudes of the accelerationpulses (a₁, a₂, a₃ . . . a_(n)) are determined via a calculationalapproximation based on the desired velocity change (V_(ref) -V₀) of thepoint of suspension, the beginning and desired final values (θ₀,θ_(ref)) of the oscillating angle and the beginning and desired finalvalues (ω₀, ω_(ref)) of the angular velocity, and the predeterminedduration of each acceleration pulse.
 11. A method for the control of aharmonically oscillating load, in which the load is transferred from abeginning state of oscillating movement of the load (θ₀, ω₀) and from abeginning velocity of a point of suspension (V₀) to a desired finalstate of oscillating movement of the load (θ_(ref), ω_(ref)) and to adesired final velocity (V_(ref)) of the point of suspension, comprisingthe steps of:(a) determining the beginning states of oscillatingmovement of the load (θ₀, ω₀) and the beginning velocity of the point ofsuspension of the load (V₀); (b) determining the desired final states ofoscillating movement of the load (θ_(ref), ω_(ref)) and the desiredfinal velocity of the point of suspension of the load (V_(ref)); (e)determining a characteristic oscillation time (τ) for the load; and (f)controlling the load with a control sequence (a(t)) comprisingconsecutively performed acceleration pulses (a_(i)), wherein the controlsequence a(t) is formed from a plurality of acceleration pulses (a₁, a₂,a₃) having a constant rate of acceleration of a calculated magnitude anda predetermined duration of τ/4, wherein the magnitudes of theacceleration pulses (a₁, a₂, a₃) are determined via a calculationalapproximation based on the desired velocity change (V_(ref) -V₀) of thepoint of suspension, the beginning and desired final values (θ₀,θ_(ref)) of the oscillating angle and the beginning and desired finalvalues (ω₀, ω_(ref)) of the angular velocity, and the predeterminedduration (τ/4), of each acceleration pulse, and wherein the magnitudesof the constant rate of acceleration differ in absolute value among theacceleration pulses.